Morse theory on spaces of braids and Lagrangian dynamics

• 成果类型: Article
• 署名作者: Ghrist, RW; Van den Berg, JB; Vandervorst, RC
• 署名单位: University of Illinois System; University of Illinois Urbana-Champaign; University of Nottingham; Vrije Universiteit Amsterdam; University System of Georgia; Georgia Institute of Technology
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-002-0277-0
• 发表日期: 2003
• 页码: 369-432
• 关键词: homotopy classes; traveling-waves; periodic-orbits; conjecture; uniqueness; index; point

摘要

In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant is derived from a Morse-Conley homotopy index. In the second half of the paper we apply this technology to second order Lagrangians via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits.